Sequential rationality, subgame perfection and backward induction in dynamic games

Sequential rationality

See Watson p. 139. A player’s strategy, which is part of a proposed strategy profile for playing the game, is sequentially rational if starting at any decision point (information set) for a player in the game (including ones that may not be reached if the game is conducted according to the strategy profile), his or her strategy from that point on represents a best response to the strategies of the other players.

A subgame

See Watson p. 141. A subgame is any part of a game tree (extensive form) that begins with an information set that has a single node (a singleton) and includes all successor nodes of that branch of the game. The game as a whole is also considered a subgame.

Subgame perfect

See Watson p. 142. A strategy profile that is a Nash equilibrium is also subgame perfect if the choices of actions specified for each node in a subgame represent a strategy profile that is a Nash equilibrium for the subgame (must be weakly congruent for that subgame). Some Nash equilibria are not subgame perfect. The concept allows the players in the game to reason that they can ignore NE that are not SGP. A concept that reduces the Nash equilibria that have to be considered in the game is called a refinement in the literature.

Illegitimate threats

Subgame perfection rules out threats of illegitimate threats. These are a claim that the player will make decisions at certain points in the game that change the responses of other players so that they avoid sending the game to those decision nodes. If these decisions are not in the interest of the player threatening them, other players will reason that the player making the threats will not carry through with them if the decision nodes are reached.

Common knowledge

Each player can examine the game from the perspective of other players and know if a strategy proposed for others is not sequentially rational.

Backward induction

With common knowledge each player can start at the end of the game and solve backwards through the different subgames to identify equilibria that are both Nash and subgame perfect.

Examples

We will be discussing a selection of examples from chapters 15, 16, and 18.